 # Primorial

For n ≥ 2, the primorial (n#) is the product of all prime numbers less than or equal to n. For example, 7# = 210 is a primorial which is the product of the first four primes multiplied together (2·3·5·7). The name is attributed to Harvey Dubner and is a portmanteau of prime and factorial. The first few primorials are:

2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, ... (sequence A002110 in OEIS) n# as a function of n (red dots), compared to n!. Both plots are logarithmic.

Notation varies; it's common to see pn#, indicating the product of the primes less than or equal to the nth prime (in other words, the product of the first n primes), and also a(n) = pn#. Asymptotically, primorials grow according to:  where "exp" is the exponential function ex and "o" is the "little-o" notation (see Big O notation). Its natural logarithm is the first Chebyshev function, written θ(n) or , which approaches the linear n for large n. pn# as a function of n, plotted logarithmically.

The idea of multiplying all primes occurs in a proof of the infinitude of the prime numbers; it is applied to show a contradiction in the idea that the primes could be finite in number.

Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, 2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 5136341251. 23# is also the common difference in arithmetic progressions of fifteen and sixteen primes.

Every highly composite number is a product of primorials (e.g. 360 = 2·6·30).

Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial n, the fraction φ(n) / n is smaller than for any lesser integer, where φ is the Euler totient function.

Any completely multiplicative function is defined by its values at primorials, since it is defined by its values at primes, which can be recovered by division of adjacent values.

Table of primorials

p p# No. of Digits
2 2 1
3 6 1
5 30 2
7 210 3
11 2310 4
13 30030 5
17 510510 6
19 9699690 7
23 223092870 9
29 6469693230 10
31 200560490130 12
37 7420738134810 13
41 304250263527210 15
43 13082761331670030 17
47 614889782588491410 18
53 32589158477190044730 20
59 1922760350154212639070 22
61 117288381359406970983270 24
67 7858321551080267055879090 25
71 557940830126698960967415390 27
73 40729680599249024150621323470 29
79 3217644767340672907899084554130 31
83 267064515689275851355624017992790 33
89 23768741896345550770650537601358310 35
97 2305567963945518424753102147331756070 37
101 232862364358497360900063316880507363070 39
103 23984823528925228172706521638692258396210 41
107 2566376117594999414479597815340071648394470 43
109 279734996817854936178276161872067809674997230 45
113 31610054640417607788145206291543662493274686990 47

* Primorial prime

References

1. ^ (sequence A002110 in OEIS)

* Harvey Dubner, "Factorial and primorial primes". J. Recr. Math., 19, 197–203, 1987. 